AP Calculus AB Unit 5 Study Guide: Analytical Applications of Differentiation

Interpreting Derivatives in Context (and What They Tell You)

A derivative is more than a rule for computing a slope. The derivative

f'(x)

measures how the output of a function

f(x)

is changing at the instant when the input is

x

In other words, it is the instantaneous rate of change of the output with respect to the input. If the function describes a real quantity (height, temperature, profit, position), then its derivative describes how fast that quantity is changing at that moment.

The derivative as slope and as rate

You will use two closely related interpretations throughout this unit.

First, the slope interpretation (geometric):

f'(a)

is the slope of the line tangent to the graph

y=f(x)

at

x=a

If the tangent line slopes upward to the right then

f'(a)>0

If it slopes downward then

f'(a)